*Forma,* Vol. 26 (No. 1), pp. 11-18, 2011

## Four-dimensional Mathematical Data Visualization via "Embodied Four-dimensional Space Display System"

Yukihito Sakai^{1,2} and Shuji Hashimoto^{3}

^{1}Faculty of Information Sciences and Arts, Toyo University, 2100 Kujirai, Kawagoe, Saitama 350-8585, Japan

^{2}Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

^{3}Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

*E-mail address: yukihito@toki.waseda.jp

(Received February 4, 2011; Accepted April 21, 2011)

**Abstract. **
In general, it is difficult to visualize 4-D mathematical data with interdependence among variables in 3-D space that we are able to perceive. In our previous work, we developed an embodied 4-D space display system for observing various 4-D objects; the system can intuitively overview and understand the shapes of 4-D objects from
various 4-D positions and directions in 4-D space. The system enables us to create any 4-D mathematical data without omitting any properties. Using the system, we can obtain various mapping structures of 4-D mathematical data such as complex functions and functions with three variables, and easily discover characteristics such as the singular point of the complex function and the extremal value of the 3-D scalar function. It is expected that our approach will be useful for the development of perception of 4-D mathematical data.

**Keywords: **
4-D mathematical data, 4-D Euclidean space, 4-D visualization, 4-D interaction

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